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Chapter 5: Problem 20

Multiply. $$ -x(x 2+x-1) $$

Short Answer

Expert verified
The expanded form is \[ -x^2 - x \].

Step by step solution

01

Distribute the Negative Sign

First, we need to distribute the eg symbol across the polynomial inside the parenthesis. Since we have a negative sign in front of the eg x, we multiply it with the first term of the polynomial eg x giving us eg x^2. After that, the product of eg x and 2 is eg 2x. Similarly, multiplying eg x with eg 1 gives us x. Therefore, the polynomial becomes eg x imes (x^2 + x - 1) = eg x^2 鈭� 2x + x.
02

Simplify the Expression

Once the distribution is complete, we need to combine any like terms in the polynomial. The terms eg 2x and x are both x terms and can be combined. Simplifying gives us eg x^2 鈭� x.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is like sharing and multiplying at the same time. It involves taking one term and multiplying it across everything inside parentheses. In this problem, you need to multiply \(-x\) by each term within the \((x^2 + x - 1)\).

Here's how it works step-by-step:
  • First, multiply \(-x\) by \(x^2\). This gives \(-x \times x^2 = -x^3\).
  • Next, multiply \(-x\) by \(x\). Here, \(-x \times x = -x^2\).
  • Lastly, multiply \(-x\) by \(-1\). Think of the negative times a negative rule, \(-x \times -1 = x\).
When you apply the distributive property, it ensures every part of the polynomial in parentheses interacts with the outside term, distributing the operation evenly.
Combining Like Terms
Combining like terms is all about organizing and simplifying expressions. "Like terms" have the same variable part, making them combinable. In our expression \(-x^3 - x^2 + x\), let's find and combine them.

Notice the coefficients:
  • The \(-x^3\) stands alone since there are no other \(x^3\) terms.
  • The terms \(-x^2\) need to be observed to see if any \(x^2\) terms exist, which they don't here.
  • The \(x\) term is by itself here.
Thus, there is minimal combining in this case beyond recognizing the structure. Knowing how to identify and mix these terms efficiently can make any polynomial multiplication much easier.
Simplifying Expressions
Simplifying expressions means making them neat and straightforward. It's like tidying up a room to see everything clearly. After applying the distributive property and combining like terms, you'll often find it necessary to simplify the expression to its most basic form.

In our expression \(-x^3 - x^2 + x\), look if you can further compress it:
  • Check if any similar terms can be combined, which in this example, they can't since they represent different powers of \(x\).
  • Verify the sign of each term and ensure correctness, especially after distributing negatives.
Ultimately, simplifying expressions means reducing the equation to its simplest possible terms, ensuring clarity and ease of understanding while avoiding unnecessary complexity.

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